Question

Make a substitution to express the integrand as a rational function and then evaluate the integral. (Use C for the constant of integration.) ∫ x ^3/ 5√x ^2 +7 ​ ​ dx

Answer 1

To evaluate the integral ∫ x ^(3/5)√(x^2 + 7) dx, we can make a substitution using u = x^(3/5) and rewrite the integral as ∫ (5/3) u^(5/2) du. Evaluating this integral using the power rule, we get (10/7) x^(21/10).

Step-by-step explanation:

To evaluate the integral ∫ x ^(3/5)√(x^2 + 7) dx, we can make a substitution to express the integrand as a rational function.

Let's substitute u = x^(3/5).

Taking the derivative of u with respect to x, we get du/dx = (3/5)x^(-2/5).

Now let's rewrite the integral using the substitution.

The numerator becomes du and the denominator becomes (x^2 + 7) dx = (5/3) u^(5/2) du.

So, the integral becomes ∫ (5/3) u^(5/2) du.

Now we can evaluate this integral using the power rule for integration. Adding one to the exponent and dividing by the new exponent, we get (10/7) u^(7/2).

Evaluating this from the limits 0 to x^(3/5), we get (10/7) (x^(3/5))^(7/2) - (10/7) (0)^(7/2).

Simplifying further, we get (10/7) x^(21/10) - 0, which simplifies to (10/7) x^(21/10).

Hence, the rational function is (10/7) x^(21/10).